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Asymptotic and non-asymptotic approaches in seismic waveform modeling and surface wave tomography

Éric Clévédé, Charles Mégnin, Barbara Romanowicz, Philippe Lognonné


Goal of this study

Surface wave data constitute a powerful tool to investigate the upper mantle structure. All the modern tomographic models of the whole mantle including these data use the same theory to model the surface waveform. This theory assumes that the lateral variation of the seismic velocities are small in amplitude and vary smoothly. Under these hyptothesis, two main approximation are done: the first-order Born approximation, where the heterogeneities are considered as secondary sources in an unperturbed reference wavefield, i.e. only single scattering is taken into account, and the asymptotic approximation, where the wavenumber of the seismic signal is much higher than the wavenumber of the structural anomalies, so the wave path can be modeled by a 0-dimension ray laterally. The goal of this study is to estimate experimentally the actual theoretical noise brought by asymptotic methods on mantle models based on surface wave tomography. To do so we perform a synthetic test using a non-asymptotic formalism, the 'Higher Order Perturbation Theory' (HOPT) (Lognonné and Romanowicz, 1990; Lognonné, 1991; Clévédé and Lognonné, 1996). This methodology incorporates the effects of back and multiple forward scattering on the wavefield by summing modes computed to third order of perturbations directly in the 3-D Earth, and models the sensitivity to scatterers away from the great-circle path. The asymptotic method used is the 'Non Linear Asymptotic Theory' (NACT) (Li and Romanowicz, 1995). This method is optimized for body wave modeling, but the surface wave modeling also offers a better accuracy in time with respect to the classical 'Path Average Approximation' (PAVA) (Woodhouse and Dziewonski, 1984).


We have designed a test model consisting in two heterogeneities with power up to spherical harmonic degree 12 (Figure 18.1). Synthetic data are computed using the HOPT method. The data set consists in 7849 seismograms corresponding to the real surface-wave data coverage used in the SAW12D model (Li and Romanowicz, 1996), in the frequency window 2.5mHz-12.5mHz (400s-80s) band-pass with a cosine taper with corner frequencies 4mHz-10mHz (125s-100s). Time windows corresponding to Love waves trains G1 and G2 are selected. We use these synthetic data as input in the inversion procedure used by Li and Romanowicz (1996), using the NACT method. Maps of the output model is shown on figure 18.2. The heterogeneities are well retreived both laterally and radially, but slighty spread in both directions, and spurious structure with small amplitude appears over all the model. The spectral content and rms profile of the models offer a better comparison: figure 18.3 show that the asymptotic filter yields a loss in energy at depths where the heterogeneities are located, and some smearing radially. In order to estimate the effect of the path coverage in this loss of resolution, we performed a 'circular' test using NACT to compute synthetic data as input. The result is shown on figure 18.4: the model is almost perfectly retrieved. We conclude that the loss of resolution shown on figure 18.3 is due to the inability of the asymptotic method to reproduce the effects of the Fresnel zone of the surface waves. We must point out that in the present case, where the initial model has only a large wavelength content, the 'theoretical noise' is still small, and that the tomographic model avalaible using surface wave data can be considered reliable, as long as they are limited to large scale structure.


The results of this study are extensively presented in
Clévédé et al. (1998) (a theoretical comparison of the NACT and HOPT methods can also be found in this paper). We have shown that we are now able to test experimentally the effect of asymptotic approximations in the framework of surface wave tomography. The next step is to investigate the limits of these approximations by using input models with a larger spectral content, and to test higher order asymptotic approximations that allow to take into account amplitude effects (focusing/defocusing) (Romanowicz, 1987). A further step will be to use the HOPT method in an iterative inversion scheme for the surface waves, coupled with the NACT method for the body waves.

Figure 18.1: Input model: the heterogeneities are located between 200 km and 600 km in depth. The model is expanded in spherical harmonics up to the degree 12 laterally and radially to the 5th-order Legendre polynomials in the upper mantle.

Figure 18.2: Output model: synthetic data are computed using the HOPT method (non-asymptotic), and inverted using the NACT method (asymptotic). Data consist in Love waves (between 400s and 80s).
\epsfig{file=figs/bsl98_eric2.eps, width=8cm}\end{center}\end{figure}

Figure 18.3: Output model: spectral content in harmonic degrees and at each depth for the input model (left) and output model (middle). The corresponding rms profile is shown on the left for the input model (solid line) and the output model (dashed line).
\epsfig{file=figs/bsl98_eric4.eps, width=8cm}\end{center}\end{figure}

Figure 18.4: Output model: same as figure 18.3 for a 'circular' test using NACT seismograms as input data.
\epsfig{file=figs/bsl98_eric3.eps, width=8cm}\end{center}\end{figure}


Clévédé, E., and P. Lognonné, Fréchet derivatives of coupled seismograms with respect to an anelastic rotating Earth, Geophys. J. Int., 124, 456-482, 1996.

Clévédé, E., C. Mégnin, B. Romanowicz and P. Lognonné, Seismic waveform modeling and surface wave tomography in a three-dimensional Earth: asymptotic and non-asymptotic approaches, Submitted to Phys. Earth. Planets Inter., 1998.

Li, X.-D., and B. Romanowicz, Comparison of global waveform inversions with and without considering cross-branch modal coupling, Geophys. J. Int., 121, 695-709, 1995.

Li, X.-D., and B. Romanowicz, Global mantle shear velocity model developed using non-linear asymptotic coupling theory, J. Geophys. Res., 101, 22245-22272, 1996.

Lognonné, P., Normal modes and seismograms of an anelastic rotating Earth, J. Geophys. Res., 96, 20309-20319, 1991.

Lognonné, P., and B. Romanowicz, Modelling of coupled normal modes of the Earth: the spectral method, Geophys. J. Int., 102, 365-395, 1990.

Romanowicz, B., Multiplet-multiplet coupling due to lateral heterogeneity: asymptotic effects on the amplitude and frequency of the Earth's normal modes, Geophys. J. R. Astr. Soc., 90, 75-100, 1987.

Woodhouse, J. H., and A. M. Dziewonski, Mapping the upper mantle: three-dimensional modeling of the Earth structure by inversion of seismic waveforms, J. Geophys. Res., 89, 5953-5986, 1984.

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Next: The effects of theoretical Up: Ongoing Research Previous: SAPSE - The Southern

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